By assumption \(H4\) , the posterior distribution is consistent at \(P\)  under the weak topology, in virtue of Schwartz theorem (see e.g.Theorem \(6.16\)  and Example \(6.20\)  in [1]}), so that
\(\text{pr}(h \in \mathcal {U}^c \mid X_{1:n}) \rightarrow 0,\) 
